Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces
© Claudio Vidal 2009
Received: 20 November 2008
Accepted: 10 June 2009
Published: 16 June 2009
The existence of periodic, almost periodic, and asymptotically almost periodic of periodic and almost periodic of abstract retarded functional difference equations in phase spaces is obtained by using stability properties of a bounded solution.
The abstract space was introduced by Hale and Kato  to study qualitative theory of functional differential equations with unbounded delay. There exists a lot of literature devoted to this subject; we refer the reader to Corduneanu and Lakshmikantham , Hino et al. . The theory of abstract retarded functional difference equations in phase space has attracted the attention of several authors in recent years. We only mention here Murakami [4, 5], Elaydi et al. , Cuevas and Pinto [7, 8], Cuevas and Vidal , and Cuevas and Del Campo .
As usual, we denote by , , and the set of all integers, the set of all nonnegative integers, and the set of all nonpositive integers, respectively. Let be the -dimensional complex Euclidean space with norm . the set .
We will denote by ( , and ) or simply by , the solution of (1.1) passing through , that is, , and the functional equation (1.1) is satisfied.
During this paper we will assume that the sequences and are bounded. The paper is organized as follows. In Section 2 we see some important implications of the fading memory spaces. Section 3 is devoted to recall definitions and some important basic results about almost periodic sequences, asymptotically almost periodic sequences, and uniformly asymptotically almost periodic functions. In Section 4 we analyze separately the cases where is periodic and when it is almost periodic. Thus, in Section 4.1 assuming that the system (1.1) is periodic and the existence of a bounded solution (particular solution) which is uniformly stable and the phase space satisfies only the axioms (A)–(C), we prove the existence of an almost periodic solution and an asymptotically almost periodic solution. If additionally the particular solution is uniformly asymptotically stable, we prove the existence of a periodic solution. Similarly, in Section 4.2 considering that system (1.1) is almost periodic and the existence of a bounded solution and whenever the phase space satisfies the axioms (A)–(C), but here it is also necessary that verifies the fading memory property. If the particular solution is asymptotically almost periodic, then system (1.1) has an almost periodic solution. While, if the particular solution is uniformly asymptotically stable, we prove the existence of an asymptotically almost periodic solution.
In [11, 12] the problem of existence of almost periodic solutions for functional difference equations is considered in the first case for the discrete Volterra equation and in the second reference for the functional difference equations with finite delay; in both cases the authors assume the existence of a bounded solution with a property of stability that gives information about the existence of an almost periodic solution. In an analogous way in  the problem of the existence of almost periodic solutions for functional difference equations with infinite delay is considered. These results can be applied to several kinds of discrete equations. However, our approach differs from Hamaya's because, firstly, in our work we consider both cases, namely, when is periodic and when it is almost periodic in the first variable. And secondly, we analyze very carefully the implications of the existence of a bounded solution of (1.1) with each property: uniformly stable, uniformly asymptotically stable, and globally uniformly stable.
Furthermore, we cite the articles [14–16] which are devoted to study almost periodic solutions of difference equations, but a little is known about almost periodic solutions, and in particular, for periodic solutions of nonlinear functional difference equations in phase space via uniform stability, uniformly asymptotically stability, and globally uniformly stability properties of a bounded solution.
2. Fading Memory Spaces and Implications
Thus, we have the following result.
3. Notations and Preliminary Results
In this section, we review the definitions of (uniformly) almost periodic, asymptotically almost periodic sequence, which have been discussed by several authors and present some related properties.
For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12].
A sequence , (or ), , equivalently, a function (or, ) is called asymptotically almost periodic if , where and (or, ) satisfying as (or, ). Denote by (or all such sequences, and is said to be an asymptotically almost periodic on (or on ) (a.a.p.) in .
For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12]. With the objective to make this manuscript self contained we decided to include the majority of the proofs.
Let be the same as in the previous lemma. Then, for any sequence , there exist a subsequence of and a function continuous in such that uniformly on as , where is any compact set in . Moreover, is also almost periodic in uniformly for .
If is a.a.p., then the decomposition , in the definition of an a.a.p. function, is unique (see ).
4. Existence of Almost Periodic Solutions
For this bounded solution , there is an such that for all . So, we will have to assume that for all , and . Next, we will point out the definitions of stability for functional difference equations adapting it from the continuous case according to Hino et al. in .
uniformly asymptotically stable, abbreviated as " '', if it is uniformly stable and there is such that for any , there is a positive integer such that if and , then for all , where is any solution of (1.1);
4.1. The Periodic Case
Before proving our following result we remark that if is a.a.p. then there are unique sequences such that , with a.p. and as as . By Lemma 3.9(a) it follows that is bounded and thus . Hence, by Axiom (C) we must have that for all . In particular, for all .
In the case when we have an asymptotically stable solution of (1.1) we obtain the following result.
By Theorem 4.5, is a.a.p. Then ), where ( ) is an a.p. sequence and as . Notice that is also a solution of (1.1) satisfying . Since is , we have that as , which implies that for all . Using same technique as in the proof of Theorem 4.7, we can show that is a -periodic solution of (1.1).
4.2. The Almost Periodic Case
and from the previous considerations the first term of the right-hand side of (4.23) tends to zero as and since as , we have that for all , which implies that (1.1) has an a.p. solution passing through , where for .
We are now in a position to prove the following result.
and hence by Axiom A(ii) for all if . This implies that the bounded solution of (1.1) is a.a.p. by Lemma 3.9(d). Furthermore, (1.1) has an a.p. solution, which is by Theorem 4.11. This ends the proof.
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